def form_identify_dof(form):
r"""Identify the DOF of a form diagram.
Parameters
----------
form : FormDiagram
The form diagram.
Returns
-------
k : int
Dimension of the null space (nullity) of the equilibrium matrix.
Number of independent states of self-stress.
m : int
Size of the left null space of the equilibrium matrix.
Number of (infenitesimal) mechanisms.
ind : list
Indices of the independent edges.
Notes
-----
The equilibrium matrix of the form diagram is
.. math::
\mathbf{E}
=
\begin{bmatrix}
\mathbf{C}_{i}^{t}\mathbf{U} \\
\mathbf{C}_{i}^{t}\mathbf{V}
\end{bmatrix}
If ``k == 0`` and ``m == 0``, the system described by the equilibrium matrix
is statically determined.
If ``k > 0`` and ``m == 0``, the system is statically indetermined with `k`
idependent states of stress.
If ``k == 0`` asnd ``m > 0``, the system is unstable, with `m` independent
mechanisms.
The dimension of a vector space (such as the null space) is the number of
vectors of a basis of that vector space. A set of vectors forms a basis of a
vector space if they are linearly independent vectors and every vector of the
space is a linear combination of this set.
Examples
--------
>>>
"""
k_i = form.key_index()
xy = form.vertices_attributes('xy')
fixed = [k_i[key] for key in form.fixed()]
free = list(set(range(len(form.vertex))) - set(fixed))
edges = [(k_i[u], k_i[v]) for u, v in form.edges()]
C = connectivity_matrix(edges)
E = equilibrium_matrix(C, xy, free)
k, m = dof(E)
ind = nonpivots(rref(E))
return k, m, [edges[i] for i in ind]
def form_identify_dof(form, **kwargs):
algo = kwargs.get('algo') or 'sympy'
k2i = form.key_index()
xyz = form.vertices_attributes('xyz')
fixed = [k2i[key] for key in form.anchors()]
free = list(set(range(form.number_of_vertices())) - set(fixed))
edges = [(k2i[u], k2i[v]) for u, v in form.edges_where({'_is_edge': True})]
C = connectivity_matrix(edges)
E = equilibrium_matrix(C, xyz, free)
return nonpivots(rref(E, algo=algo))
